204
IX – Multivariate Differential and Integral Calculus
This formula can be made to apply to much more general situations:
Theorem 4. Let U ⊂ R
n be an open bounded set, A its closure and ϕ a
diffeomorphism of class C
1 from A to a compact subset B ⊂ R
n . Suppose
that the borders of A and B have measure zero. Then, for any integrable
39
function f on B,
ϕ(A)
f (x)dx =
A
f [ϕ(t)] . |J ϕ (t)| dt
(10.4)
denoting by x = (x
1 , . . . , x
n ) or t = (t
1 , . . . , t
n ) the integration variable, by
dm(x) = dx = dx
1 . . . dx
n or dt the Lebesgue measure on R
n and writing
for a multiple integral. This is similar to what happens in a non-oriented
simple integral in which a strictly monotone change of variable x = ϕ(t) is
being carried out: the factor ϕ
(t) involved in the oriented integrals has to be
replaced by its absolute value:
ϕ(I)
f (x)dx =
I
f [ϕ(t)] . |ϕ
(t)| dt
since otherwise the sides could have opposite signs.
We will prove the formula for continuous functions. By n
◦ 10 and 11 of
Chapter V, § 2 it can then be immediately generalized to lsc (resp. usc) functions; for this, simply observe that if there is an increasing (resp. decreasing)
philtre Φ for continuous functions, then the functions f [ϕ(s, t)]|J ϕ (s, t)|,where
f ∈ Φ, form an increasing (resp. decreasing) philtre of continuous functions
on A . Hence it is possible to pass to the limit under the
sign on both sides
of this formula. Lebesgue’s theorems lead to the general case in a few lines.
In particular, it applies if f is the characteristic function of an open or closed
subset of ϕ(A), hence of the form ϕ(M ) where M ⊂ A is an open or closed
subset in A. We find the measure of ϕ(M ) on the left hand side; on the right,
we integrate the characteristic function of M , whence
m [ϕ(M )] =
M
|J ϕ (t)| dt .
(10.4’)
(i) Case where ϕ is linear. This is the simplest case and we will give
a proof using classical properties
40 of the group GL n (R) = G of n × n real
39 In the sense of Lebesgue.
40 As shown by K. Iwasawa about 1950 in an article rightly famous, in particular lemma d below if it is properly interpreted, these properties generalize to
all semisimple Lie groups, a class of groups singled out by ´
Elie Cartan whose
extraordinary properties continue to be the subject of numerous studies mixing algebraic geometry, number theory, generalizations of the theory of modular
functions, non-commutative harmonic analysis (there is a quite sophisticated version of Fourier transforms for these groups), PDEs, etc. Apart from “ classical ”
groups such as the matrix group equipped with a symmetric or alternating bi-
IX – Multivariate Differential and Integral Calculus
This formula can be made to apply to much more general situations:
Theorem 4. Let U ⊂ R
n be an open bounded set, A its closure and ϕ a
diffeomorphism of class C
1 from A to a compact subset B ⊂ R
n . Suppose
that the borders of A and B have measure zero. Then, for any integrable
39
function f on B,
ϕ(A)
f (x)dx =
A
f [ϕ(t)] . |J ϕ (t)| dt
(10.4)
denoting by x = (x
1 , . . . , x
n ) or t = (t
1 , . . . , t
n ) the integration variable, by
dm(x) = dx = dx
1 . . . dx
n or dt the Lebesgue measure on R
n and writing
for a multiple integral. This is similar to what happens in a non-oriented
simple integral in which a strictly monotone change of variable x = ϕ(t) is
being carried out: the factor ϕ
(t) involved in the oriented integrals has to be
replaced by its absolute value:
ϕ(I)
f (x)dx =
I
f [ϕ(t)] . |ϕ
(t)| dt
since otherwise the sides could have opposite signs.
We will prove the formula for continuous functions. By n
◦ 10 and 11 of
Chapter V, § 2 it can then be immediately generalized to lsc (resp. usc) functions; for this, simply observe that if there is an increasing (resp. decreasing)
philtre Φ for continuous functions, then the functions f [ϕ(s, t)]|J ϕ (s, t)|,where
f ∈ Φ, form an increasing (resp. decreasing) philtre of continuous functions
on A . Hence it is possible to pass to the limit under the
sign on both sides
of this formula. Lebesgue’s theorems lead to the general case in a few lines.
In particular, it applies if f is the characteristic function of an open or closed
subset of ϕ(A), hence of the form ϕ(M ) where M ⊂ A is an open or closed
subset in A. We find the measure of ϕ(M ) on the left hand side; on the right,
we integrate the characteristic function of M , whence
m [ϕ(M )] =
M
|J ϕ (t)| dt .
(10.4’)
(i) Case where ϕ is linear. This is the simplest case and we will give
a proof using classical properties
40 of the group GL n (R) = G of n × n real
39 In the sense of Lebesgue.
40 As shown by K. Iwasawa about 1950 in an article rightly famous, in particular lemma d below if it is properly interpreted, these properties generalize to
all semisimple Lie groups, a class of groups singled out by ´
Elie Cartan whose
extraordinary properties continue to be the subject of numerous studies mixing algebraic geometry, number theory, generalizations of the theory of modular
functions, non-commutative harmonic analysis (there is a quite sophisticated version of Fourier transforms for these groups), PDEs, etc. Apart from “ classical ”
groups such as the matrix group equipped with a symmetric or alternating bi-